
In optimizing \(\phi \)  and the hyperparameters \(\sigma ^2\)  at the same time we here follow a common scheme, cf. [1]}. To solve (REF ), we use backpropagation on the following Monte-Carlo object
\(\begin{aligned}\mathcal {L}^{\mathrm {M.C.}}(\phi ; \sigma ^2) &:= -\frac{1}{M} \sum _{m=1}^M \log \left(\frac{1}{L} \sum _{l=1}^L p(y_{i_m}|\theta _m, \zeta _{i_m,l}, \sigma _y^2 ) \right) \\&+ \frac{1}{N} D_{\mathrm {KL}}(q_{\phi }(\theta )\Vert \pi (\theta ))-\frac{1}{M}\sum _{m=1}^{M}\log \pi (x_{i_m}|\sigma _x^2) \,,\end{aligned}\) 
